A question regarding state spaces

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Discussion Overview

The discussion revolves around understanding state spaces in quantum mechanics, particularly focusing on generalized function bases, Dirac notation, and the transition from finite-dimensional spaces to Hilbert spaces. Participants seek resources and clarification on these concepts, which involve both theoretical and mathematical aspects.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses a need for resources that explain generalized function bases and their relation to Dirac notation in an understandable way.
  • Another participant corrects a spelling issue regarding the plural of "basis," indicating a focus on clarity in communication.
  • A participant acknowledges the complexity of their original question and seeks more precise guidance on transitioning from finite-dimensional spaces to Hilbert spaces.
  • There is a suggestion to refer to standard textbooks on functional analysis, specifically mentioning Reed + Simon and Thirring, for rigorous mathematical treatment of Hilbert spaces.
  • One participant shares a personal webpage they created to provide a straightforward description of quantum mechanics state spaces.
  • Another participant recommends a functional analysis book by Erwin Kreyszig that covers Hilbert and Banach spaces.

Areas of Agreement / Disagreement

Participants generally agree on the need for better resources and clarification regarding the mathematical transition to Hilbert spaces. However, there is no consensus on specific resources or the best approach to understanding these concepts, as multiple suggestions and viewpoints are presented.

Contextual Notes

Some participants express uncertainty about the clarity of their questions and the adequacy of existing literature, indicating potential limitations in the resources available for understanding the transition from finite to infinite-dimensional spaces.

Who May Find This Useful

Individuals interested in quantum mechanics, functional analysis, and the mathematical foundations of Hilbert spaces may find this discussion beneficial.

vacuum
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I am trying to get oriented in all those QM spaces.

I would appreciate if somenone could point out some source of information (preferably online, though paper is also fine :-) ) which explains the matter in form understandable to humans.

More specifically I need explanation of generalised functions basises (Latin plural of basis would be 'basii'?) and representation of state functions in them as well as how does the Dirac notation transcend to these new objects (As far as I know it was defined with the aid of space - dual space product, which is somewhat more complicated when talking about rigged Hilbert spaces etc.)
 
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Originally posted by vacuum
(Latin plural of basis would be 'basii'?)

plural in latin and in english (check your dictionary next time you need to know a spelling, or if you don t have one, use dictionary.com) of the word "basis" is "bases"
 
Thank you very much for resolving this issue Lethe.

However those other questions still remain. Perhaps I should post this in some other section? Any hints?
 
Originally posted by vacuum
Thank you very much for resolving this issue Lethe.

However those other questions still remain. Perhaps I should post this in some other section? Any hints?
i m not quite sure what your question is, which is why i didn t say anything. if you want to learn about hilbert spaces, you can probably start with a beginning quantum mechanics book. or if you are more serious, a functional analysis book.
 
Originally posted by lethe
i m not quite sure what your question is, which is why i didn t say anything. if you want to learn about hilbert spaces, you can probably start with a beginning quantum mechanics book. or if you are more serious, a functional analysis book.

Yes, when I re-read my original post I found it rather uncomprehensible. I wanted to put too much questions in too little space.

Let me try to explain what I need more precisely:

I had a course of mathematical physics in which (for example) a Dirac notation was introduced and this course was based finite dimension spaces.

Now, when I got to Hilbert spaces there was a leap without much mathematical rigour from finite spaces to Hilbert spaces using the same mathematical formalism! I know now how to employ those mathematic instruments to solve problems, but the justification for using them was left out.

Since I didn't find any sort of strict mathematical transition from one case to another in literature I have on disposal, I was wondering if anyone here can recommend me something (more specific than 'functional analysis book' :)
 
Originally posted by vacuum
Since I didn't find any sort of strict mathematical transition from one case to another in literature I have on disposal, I was wondering if anyone here can recommend me something (more specific than 'functional analysis book' :)

are there any specific issues you would like to ask questions about?

anyway, a standard textbook for functional analysis is Reed + Simon, Functional Analysis, part of his series on mathematical physics. also try Quantum Mathematical Physics by Thirring, and Quantum Field Theory by Ticciati. those books also deal with the mathematics of Hilbert spaces rather rigorously.

i will retract my earlier advice that you check a beginning quantum mechanics book. those books tend to gloss over at best, or ignore entirely at worst, the mathematical problems that arise when you move from finite dimensional to infinite dimensional Hilbert spaces.
 
Last edited:
Originally posted by vacuum
I am trying to get oriented in all those QM spaces.

I would appreciate if somenone could point out some source of information (preferably online, though paper is also fine :-) ) which explains the matter in form understandable to humans.

I was refreshing myself this past summer on this and thought it would be nice to make a web page to have a clear straightforward description of all this. Here is the page I created --
http://www.geocities.com/physics_world/qm/state_space.htm

Hope it helps
 
Thanks people. This was very helpful for me!
 
hi!
there is a good functional analysis book by Erwin Kreyzig...which has hilbert spaces and banach spaces and stuff!
 

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